CS 2710, ISSP 2160

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CS 2710, ISSP 2160

• The Situation Calculus
• KR and Planning
• Some final topics in KR

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Situation Calculus

• Planning in propositional logic Section 7.7
  through 7.7.2
• Section 10.4.2
• Handouts
• Note Fall 2015 you are only responsible for
  some of these notes, specifically what we covered
  11/3/2015

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Other topics in KR

• Semantic Networks 12.5.1
• Description Logic 12.5.2 we didnt cover this
• Satisfiability and WalkSat Intro to Section
  7.6 7.6.2 we didnt cover this

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Actions, Situations, and EventsThe Situation
Calculus

• The robot is in the kitchen.
• in(robot,kitchen)
• He walks into the living room.
• in(robot,livingRoom)
• in(robot,kitchen,202pm)
• in(robot,livingRoom,217pm)
• But what if you are not sure when it was?
• We can do something simpler than rely on time
  stamps
• The Situation Calculus is a logic formalism for
  representing and reasoning about dynamic domains.

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Situation Calculus Ontology

• Actions terms, such as forward and
  turn(right))
• Situations terms initial situation, say s0,
  and all situations that are generated by applying
  an action to a situation. result(a,s) names the
  situation resulting when action a is done in
  situation s.

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Situation Calculus Ontology continued

• Fluents functions and predicates that vary from
  one situation to the next. By convention, the
  situation is the last argument of the fluent.
  holding(robot,gold,s0)
• Atemporal or eternal predicates and functions do
  not change from situation to situation.
  gold(g1). lastName(wumpus,smith).
  adjacent(livingRoom,kitchen).

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Sequences of Actions

• Also useful to reason about action sequences
• All S resultSeq(,S) S
• All A,Se,S resultSeq(ASe,S)
  resultSeq(Se,result(A,S))
• resultSeq(a,b,c,so) is
• result(c,result(b,result(a,s0)

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Modified Wumpus World

• Fluent predicates at(O,X,S) and holding(O,S)
• In our simple world, only the agent can hold a
  piece of gold, so for simplicity, only the gold
  and situation are arguments
• Initial situation at(agent,1,1,s0)
  at(g1,1,2,s0)
• But we want to exclude possibilities from the
  initial situation too

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Initial KB

• All O,X (at(O,X,s0) ?? (Oagent X 1,1) v
  (Og1 X 1,2))
• All O holding(O,s0)
• Eternals
• gold(g1) adjacent(1,1,1,2)
  adjacent(1,2,1,1) etc.

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Goal g1 is in 1,1

• Planning by answering the query
• Exists S at(g1,1,1,resultSeq(S,s0))
• Solution
• At(g1,1,1,resultSeq(go(1,1,1,2),grab(g1),go
  (1,2,1,1),s0))
• The situation designated by the second term
• result(go(1,1,1,2),result(grab(g1),result(go(
  1,1,1,2),s0)))
• Lets look at what has to go in the KB for such
  queries to be answered...

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Possibility and Effect Axioms

• Possibility axioms
• Preconditions ? poss(A,S)
• Effect axioms
• poss(A,S) ? changes that result from that action

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Axioms for our Wumpus World

• For brevity we will omit universal quantifies
  that range over entire sentence. S ranges over
  situations, A ranges over actions, O over objects
  (including agents), G over gold, and X,Y,Z over
  locations.

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Possibility Axioms

• The possibility axioms that an agent can
• go between adjacent locations,
• grab a piece of gold in the current location, and
  
• release gold it is holding

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Effect Axioms

• If an action is possible, then certain fluents
  will hold in the situation that results from
  executing the action
• Going from X to Y results in being at Y
• Grabbing the gold results in holding the gold
• Releasing the gold results in not holding it

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Frame Problem

• We run into the frame problem
• Effect axioms say what changes, but dont say
  what stays the same
• A real problem, because (in a non-toy domain),
  each action affects only a tiny fraction of all
  fluents

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Frame Problem (continued)

• One solution approach is writing explicit frame
  axioms, such as
• (at(O,X,S) (Oagent) holding(O,S)) ?
  at(O,X,result(Go(Y,Z),S))
• If something is at X in S, and it is not the
  agent, and also it is not something the agent
  holds, then O is still at X if the agent moves
  somewhere.
• F fluents and A actions O(FA) axioms needed
• We can do something more efficient than this

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Frame Problem

• What stays the same?
• A actions, F fluents, and E effects/action (worst
  case). Typically, E ltlt F
• That is, the effects of an action are typically
  only a small set of all the things that could
  change
• Want O(AE) versus O(AF) solution

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Solving the Frame Problem

• For each fluent, have successor-state axioms
• Action is possible ?
• (fluent is true in result state ??
• actions effect made it true v
• it was true before and action left it alone)
• Each of the E effects of each of the A actions is
  mentioned exactly once, so O(AE) axioms needed
• Note we will return to this point later, after
  going through the wumpus world example

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Initial KB (reminder)

• All O,X at(O,S,s0) ?? Oagent X 1,1) v
  (Og1 X 1,2)
• All O holding(O,s0)
• Eternals
• gold(g1) adjacent(1,1,1,2)
  adjacent(1,2,1,1).
• Trace through reasoning so far on board
• state space handed out
• At this point, we are switching to variables
  being small case, constants upper case, following
  the text

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4-5 are Successor-State Axioms

• At(Agent, x, s) ? Adjacent(x,y) ? Poss(Go(x,y),s)
• Gold(g) ? At(Agent,x,s) ? At(g, x, s) ?
  Poss(Grab(g),s)
• Holding(g,s) ? Poss(Release(g),s)
• Poss(a,s) ? Holding(g,Result(a,s)) ?
  a Grab(g) v
  (Holding(g,s) ? a ? Release(g)))
• Poss(a,s) ? (At(o,y,Result(a,s)) ? (a Go(x,y)
  ? (o Agent v Holding(o,s)))
  v (At(o,y,s) ? (?z y ? z ? a Go(y,z) ?
  (o Agent v
  Holding(o,s))))

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More explicit version Replaced existential with
universal in 5

• All x,y,s ((At(Agent, x, s) ? Adjacent(x,y)) ?
  Poss(Go(x,y),s)
• All g,x,s ((Gold(g) ? At(Agent,x,s) ? At(g, x,
  s)) ? Poss (Grab(g),s))
• All g,s (Holding(g,s) ? Poss(Release(g),s))
• All a,s,g (Poss(a,s) ? (Holding(g,Result(a,s)) ?
  (a
  Grab(g) v (Holding(g,s) ? a ? Release(g))))
• All a,s,o,y,z (Poss(a,s) ? (At(o,y,Result(a,s))
  ? ((a Go(x,y) ? (o Agent v Holding(o,s)))
  v (At(o,y,s) ? (a
  Go(y,z) ? y ? z ?
  (o Agent v Holding(o,s)))))))
• Justification previous 5 has (?z the change
  is justified because this is equivalent to all z
  

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Same as previous, but without comments

1. All x,y,s ((At(Agent, x, s) ? Adjacent(x,y)) ?
   Poss(Go(x,y),s)
2. All g,x,s ((Gold(g) ? At(Agent,x,s) ? At(g, x,
   s)) ? Poss (Grab(g),s))
3. All g,s (Holding(g,s) ? Poss(Release(g),s))
4. All a,s,g (Poss(a,s) ? (Holding(g,Result(a,s)) ?
   (a
   Grab(g) v (Holding(g,s) ? a ? Release(g))))
5. All a,s,o,y,z (Poss(a,s) ? (At(o,y,Result(a,s))
   ? ((a Go(x,y) ? (o Agent v Holding(o,s)))
   v (At(o,y,s) ? (a
   Go(y,z) ? y ? z ?
   (o Agent v Holding(o,s)))))))

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A return to complexity

• Each of the E effects of each of the A actions is
  mentioned exactly once, so O(AE) axioms needed
• Notes
• an effect may be to make a fluent true (add it)
  or to make it false (delete it)
• Counting axioms is a bit arbitrary, since a
  single axiom may mention a disjunction of add
  effects and/or a disjunction of delete effects
  (see the holding axiom for the blocks world)
• It is true that each of the add or delete effects
  of each action is mentioned once

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A return to complexity

• Each of the E effects of each of the A actions is
  mentioned exactly once, so O(AE) axioms needed
  total action mentions
• Another schema for successor state axioms
• Action is possible ?
• (fluent is true in result state ??
• (action is one of the actions that makes it
  true (add effect) v
• it was true before and action is not one of
  the actions that makes it false (delete effect))
• Thus, for each fluent (possible effect), we
  mention exactly once each action that has it as
  an effect (either making it true or making it
  false)
• Or, for each action, it is mentioned exactly once
  in a successor state axiom for each of its
  effects

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Fall 2014

• In class exercise the blocks world handout

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Qualification Problem

• Ensuring that all necessary conditions for an
  actions success have been specified. No
  complete solution in logic. KR/planning
  designers have to decide how much detail to go
  into.

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What did we see?

• A sophisticated KR scheme
• Important Problems in planning
• Addressed by successor-state axioms (Reiter 1991)
• Frame Problem (what stays the same?)
• Ramification Problem (implicit effects, such as
  that gold moves too if the agent moves and it is
  holding the gold)
• Not addressed completely in logic
• Qualification Problem
• Concepts for planning, such as fluents and
  situations
• Planning as search

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Stopped here, Fall 2014
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Semantic Networks

• Graphical aids for visualizing the knowledge base
• Efficient algorithms for inferring properties
  based on category membership
• Often, correspond to a subset of first-order
  logic
• Many variants
• All distinguish among individual objects,
  categories of objects and relations among objects

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Example

• See figure 12.5 (next slide)
• Specify what edges and nodes mean
• In Figure 12.5, indivs and categories look the
  same
• memberOf(indiv,category)
• sisterOf(indiv,indiv)
• subsetOf(category,category)
• hasMother(indiv,indiv)

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Semantic Networks

• Is hasMother(persons,femalePersons) consistent
  with the representation?
• Nope hasMother is a relation between
  individuals
• cat1-- label ? cat2 means
• all X (X in cat1 ? (all Y label(X,Y) ? Y in
  cat2))
• (Note this does not say that each person has a
  mother)

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Semantic Networks

• cat label ? value
• All X (X in cat ? label(X,value))

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Inheritance

• Inheritance is efficient and convenient
• Trace paths from individuals to categories,
  inheriting properties as you go
• In Figure 12.5, how many legs does John have?
  Most specific (nearest) information wins

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Semantic Networks

• In this type of semantic network, only binary
  relations are possible
• A richer representation is possible by reifying
  propositions and events (example SNePS)
• This forces creation of a rich ontology of
  reified concepts many current ideas originated
  in semantic network systems

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Description Logics

• http//en.wikipedia.org/wiki/Description_logic
• This wont be tested on the exam, but I want you
  to know what description logics are
• Subset of full first order logic a family of
  logics of increasing expressiveness well
  studied most are decidable good link between
  theory and practice.

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• We stopped here Fall 2012.

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Proof methods

• Proof methods divide into (roughly) two kinds
• Application of inference rules
• Legitimate (sound) generation of new sentences
  from old.
• Resolution
• Forward Backward chaining
• Model checking
• Searching through truth assignments.
• Improved backtracking Davis--Putnam-Logemann-Love
  land (DPLL)
• Heuristic search in model space Walksat.

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Model Checking

• Two families of efficient algorithms
• Complete backtracking search algorithms DPLL
  algorithm. You read this on your own for the
  midterm.
• Incomplete local search algorithms
• WalkSAT algorithm

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The DPLL algorithm

• Determine if an input propositional logic
  sentence (in CNF) is
• satisfiable. This is just backtracking search for
  a CSP.
• Improvements
• Early termination
• A clause is true if any literal is true.
• A sentence is false if any clause is false.
• Pure symbol heuristic
• Pure symbol always appears with the same "sign"
  in all clauses.
• e.g., In the three clauses (A ? ?B), (?B ? ?C),
  (C ? A), A and B are pure, C is impure.
• Make a pure symbol literal true
• 3 Unit clause heuristic
• Unit clause only one literal in the clause
• The only literal in a unit clause must be true.
• Note literals can become a pure symbol or a
• unit clause when other literals obtain truth
  values. e.g.

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The WalkSAT algorithm

• Incomplete, local search algorithm
• Evaluation function The min-conflict heuristic
  of minimizing the number of unsatisfied clauses
• Balance between greediness and randomness
• See figure 7.18 (on your own)

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WrapUp Situation Calculus

• Planning in propositional logic Section 7.7
  through 7.7.2
• Section 10.4.2
• Handouts

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WrapUp Other topics in KR

• Semantic Networks 12.5.1 not covered Fall
  2014